The “Dog Running Between Two People” Problem

“Jim & Jan, 600 feet apart, walk toward each other. Jim walks at 100 ft/min, while Jan walks 50 ft/min. Their dog is standing next to Jim, and when Jim starts to walk toward Jan, the dog runs toward Jan at 300 ft/min, then turns around and returns to Jim, and so forth until all three meet. Approximately how many feet does the dog run?”

The first time you encounter a question like this, it sounds like it involves calculus, or at least a lot of algebra. The GMAT loves including questions like this on its test.

One of the common question types on the GMAT is the “rate problem”: you are given rates for people moving or completing a task. When two people are both doing the same “job” — for instance, walking — you can simply add their rates together to find out their combined speed.

In this case, Jim and Jan are traveling toward each other at a combined rate of 100 ft/min + 50 ft/min = 150 ft/min. You can use the Distance formula (D = rt) to calculate that the time they will talk is equal to the Distance of 600 feet divided by their combined rate of 150 ft/sec, or 4 minutes. They won’t meet exactly in the middle — Jim will travel 400 feet, while Jan will travel 200 feet — but we don’t need to calculate this to solve the problem.

Calculating the actual distance the dog traveled would be difficult using calculus, and would require using functions and limits. However, if you know that the dog will be traveling at a constant rate of 300 feet per second until they meet, it doesn’t matter in what pattern he runs; he will be running for four minutes exactly, and in those two minutes he will travel 1,200 feet. (Ignoring any time lost for the dog turning around to change direction.)

The moral to this story: when on a test like the GMAT, don’t fall for the test’s attempt to get you to use difficult calculations. Look for a simple way to solve the problem; virtually all the problems can be solved in under 3 minutes with one of several basic strategies.

If you enjoy math problems like this, you would probably enjoy, and do well on, the GMAT test. Contact Bobby Hood Test Prep to discuss the various class and tutoring options for the GMAT, both locally in Austin and online through The Princeton Review’s LiveOnline classrooms.